Independence of multipliers in several variables complex dynamics

We establish the independence of multipliers for polynomial endomorphisms of ${\mathbb{C}}^n$ and endomorphisms of ${\mathbb{P}}^n\mathrm{.}$ This allows us to extend results about the bifurcation measure and the critical height obtained in \cite{arXiv:2305.02246} to the case of polynomial endomorphisms of ${\mathbb{C}}^n$ for $n\ge 3$. An important step in the proof is the irreducibility of the spaces of endomorphisms with $N$ marked periodic points, which is of independent interest.